40.17.7 problem 17

Internal problem ID [6808]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 26. Integration in series (singular points). Supplemetary problems. Page 218
Problem number : 17
Date solved : Sunday, March 30, 2025 at 11:23:22 AM
CAS classification : [_Lienard]

\begin{align*} x y^{\prime \prime }+2 y^{\prime }+x y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 32

Order:=6; 
ode:=x*diff(diff(y(x),x),x)+2*diff(y(x),x)+x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \left (1-\frac {1}{6} x^{2}+\frac {1}{120} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 42

ode=x*D[y[x],{x,2}]+2*D[y[x],x]+x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {x^3}{24}-\frac {x}{2}+\frac {1}{x}\right )+c_2 \left (\frac {x^4}{120}-\frac {x^2}{6}+1\right ) \]

✓ Sympy. Time used: 0.990 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*y(x) + x*Derivative(y(x), (x, 2)) + 2*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{120} - \frac {x^{2}}{6} + 1\right ) + \frac {C_{1} \left (- \frac {x^{6}}{720} + \frac {x^{4}}{24} - \frac {x^{2}}{2} + 1\right )}{x} + O\left (x^{6}\right ) \]